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Question

Two positive integers are chosen. Their sum is revealed to perfect logician A. The sum of their squares is revealed to perfect logician B. This is all common knowledge. The following conversation ensues:

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: Now I know what the two numbers are!

What are the two numbers?

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First, I must say I like this puzzle very much. It's based on pure logic and math.

At first you might think that Logician B should know the answer because you might think that the sum of the squares of two positive integers is unique. But not all of them are unique. For example, if the number revealed to B is 20, he would immediately know that the two numbers are 2 & 4 for there is no other solution. But since he didn't know in the first place, then the two numbers will be for example 2,8 or 3,7 or 1,7 or 5,5. I didn't search for higher alternatives just to simplify the puzzle for lazy me.

When B said I don't know, A thought about the solutions above and and tried to match one of them with the sum he has. For example, he knows the sum is 10. There are still two options: 2,8 & 3,7. Therefore he still doesn't know.

After this point I'm stuck. I can't imagine how possible it is to know the answer when having two perfect solutions.

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Two positive integers are chosen. Their sum is revealed to perfect logician A. The sum of their squares is revealed to perfect logician B. This is all common knowledge. The following conversation ensues:

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: I don’t know what the two numbers are.

Logician A: I don’t know what the two numbers are.

Logician B: Now I know what the two numbers are!

What are the two numbers?

Logician A knows x + y

Logician B knows x^2 + y^2

For simplicity, assume x <= y

Because Logician B does not know immediately, x^2 + y^2 must not be a unique sum of squares.

Here's some numbers to play with:

(1,8) and (4,7) give 65

(2,9) and (6,7) give 85

(2,11) and (5,10) give 125

(3,11) and (7,9) give 130

(1,12) and (8,9) give 145

Let's flip this around and look at x+y

In the same "grid", here are the sums

9, 11

11, 13

13, 15

14, 16

13, 17

If logician A's sum was 13

the three possible number pairs are

(6,7), (2,11), (1,12)

If his sum were 11, then the possible pairs would be

(4,7) and (2,9)

We are now at line 4 in the riddle,

Back to logician B.

Logician B must know that logician A cannot figure out the number despite the logic I showed above.

Logician B still cannot figure it out though.

This means that x^2+y^2 must have pairs common to both the 13 case and the 11 case above.

The only number we have considered so far that does this is 85 which gives pairs (6,7) and (2,9)

Logician A knows B cannot figure it out, but here we run into a problem, A now knows the answer.

If A's sum were 11, he would say the numbers were (6,7), if his sum were 13, he would say the numbers were (2,9).

To go to yet another round, there would need to be another pair of pairs of numbers that are common to cases that A could have.

We need to get more numbers

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Logician A knows x + y

Logician B knows x^2 + y^2

For simplicity, assume x <= y

Because Logician B does not know immediately, x^2 + y^2 must not be a unique sum of squares.

Here's some numbers to play with:

(1,8) and (4,7) give 65

(2,9) and (6,7) give 85

(2,11) and (5,10) give 125

(3,11) and (7,9) give 130

(1,12) and (8,9) give 145

Let's flip this around and look at x+y

In the same "grid", here are the sums

9, 11

11, 13

13, 15

14, 16

13, 17

If logician A's sum was 13

the three possible number pairs are

(6,7), (2,11), (1,12)

If his sum were 11, then the possible pairs would be

(4,7) and (2,9)

We are now at line 4 in the riddle,

Back to logician B.

Logician B must know that logician A cannot figure out the number despite the logic I showed above.

Logician B still cannot figure it out though.

This means that x^2+y^2 must have pairs common to both the 13 case and the 11 case above.

The only number we have considered so far that does this is 85 which gives pairs (6,7) and (2,9)

Logician A knows B cannot figure it out, but here we run into a problem, A now knows the answer.

If A's sum were 11, he would say the numbers were (6,7), if his sum were 13, he would say the numbers were (2,9).

To go to yet another round, there would need to be another pair of pairs of numbers that are common to cases that A could have.

We need to get more numbers

I have to leave, but here is a big list of numbers to solve the problem.

I didn't finish separating them on different lines

Someone interested can copy and paste and then reseperate, or reparse, or whatever.

Good luck.

Here is what the meaning is:

[x^2+y^2, x+y, x, y]

[65, 9, 1, 8], [65, 11, 4, 7],

[85, 11, 2, 9], [85, 13, 6, 7],

[125, 13, 2, 11], [125, 15, 5, 10],

[130, 14, 3, 11], [130, 16, 7, 9],

[145, 13, 1, 12], [145, 17, 8, 9],

[170, 14, 1, 13], [170, 18, 7, 11],

[185, 17, 4, 13], [185, 19, 8, 11],

[205, 17, 3, 14], [205, 19, 6, 13],

[221, 19, 5, 14], [221, 21, 10, 11],

[250, 20, 5, 15], [250, 22, 9, 13],

[260, 18, 2, 16], [260, 22, 8, 14],

[265, 19, 3, 16], [265, 23, 11, 12],

[290, 18, 1, 17], [290, 24, 11, 13],

[305, 21, 4, 17], [305, 23, 7, 16],

[325, 19, 1, 18], [325, 23, 6, 17], [325, 25, 10, 15],

[340, 22, 4, 18], [340, 26, 12, 14],

[365, 21, 2, 19], [365, 27, 13, 14],

[370, 22, 3, 19], [370, 26, 9, 17],

[377, 23, 4, 19], [377, 27, 11, 16],

[410, 26, 7, 19], [410, 28, 11, 17],

[425, 25, 5, 20], [425, 27, 8, 19], [425, 29, 13, 16],

[442, 22, 1, 21], [442, 28, 9, 19],

[445, 23, 2, 21], [445, 29, 11, 18],

[481, 29, 9, 20], [481, 31, 15, 16],

[485, 23, 1, 22], [485, 31, 14, 17],

[493, 25, 3, 22], [493, 31, 13, 18],

[500, 26, 4, 22], [500, 30, 10, 20],

[505, 29, 8, 21], [505, 31, 12, 19],

[520, 28, 6, 22], [520, 32, 14, 18],

[530, 24, 1, 23], [530, 32, 13, 19],

[533, 25, 2, 23], [533, 29, 7, 22], [545, 27, 4, 23], [545, 33, 16, 17], [565, 29, 6, 23], [565, 31, 9, 22], [580, 26, 2, 24], [580, 34, 16, 18], [585, 27, 3, 24], [585, 33, 12, 21], [610, 32, 9, 23], [610, 34, 13, 21], [625, 31, 7, 24], [625, 35, 15, 20], [629, 27, 2, 25], [629, 33, 10, 23], [650, 30, 5, 25], [650, 34, 11, 23], [650, 36, 17, 19], [680, 28, 2, 26], [680, 36, 14, 22], [685, 29, 3, 26], [685, 37, 18, 19], [689, 33, 8, 25], [689, 37, 17, 20], [697, 35, 11, 24], [697, 37, 16, 21], [725, 33, 7, 26], [725, 35, 10, 25], [725, 37, 14, 23], [730, 28, 1, 27], [730, 38, 17, 21], [740, 34, 8, 26], [740, 38, 16, 22], [745, 31, 4, 27], [745, 37, 13, 24], [754, 32, 5, 27], [754, 38, 15, 23], [765, 33, 6, 27], [765, 39, 18, 21], [785, 29, 1, 28], [785, 39, 16, 23], [793, 31, 3, 28], [793, 35, 8, 27], [820, 34, 6, 28], [820, 38, 12, 26], [845, 31, 2, 29], [845, 39, 13, 26], [845, 41, 19, 22], [850, 32, 3, 29], [850, 38, 11, 27], [850, 40, 15, 25], [865, 37, 9, 28], [865, 41, 17, 24], [884, 38, 10, 28], [884, 42, 20, 22], [890, 36, 7, 29], [890, 42, 19, 23], [901, 31, 1, 30], [901, 41, 15, 26], [905, 37, 8, 29], [905, 39, 11, 28], [925, 35, 5, 30], [925, 41, 14, 27], [925, 43, 21, 22], [949, 37, 7, 30], [949, 43, 18, 25], [962, 32, 1, 31], [962, 40, 11, 29], [965, 33, 2, 31], [965, 43, 17, 26], [970, 34, 3, 31], [970, 44, 21, 23], [985, 41, 12, 29], [985, 43, 16, 27], [986, 36, 5, 31], [986, 44, 19, 25], [1000, 40, 10, 30], [1000, 44, 18, 26], [1010, 38, 7, 31], [1010, 42, 13, 29], [1025, 33, 1, 32], [1025, 39, 8, 31], [1025, 45, 20, 25], [1037, 43, 14, 29], [1037, 45, 19, 26], [1040, 36, 4, 32], [1040, 44, 16, 28], [1060, 38, 6, 32], [1060, 46, 22, 24], [1066, 44, 15, 29], [1066, 46, 21, 25], [1073, 39, 7, 32], [1073, 45, 17, 28], [1090, 34, 1, 33], [1090, 46, 19, 27], [1105, 37, 4, 33], [1105, 41, 9, 32], [1105, 43, 12, 31], [1105, 47, 23, 24], [1125, 39, 6, 33], [1125, 45, 15, 30], [1130, 44, 13, 31], [1130, 46, 17, 29], [1145, 43, 11, 32], [1145, 47, 19, 28], [1157, 35, 1, 34], [1157, 45, 14, 31], [1160, 36, 2, 34], [1160, 48, 22, 26], [1165, 37, 3, 34], [1165, 47, 18, 29], [1170, 42, 9, 33], [1170, 48, 21, 27], [1189, 43, 10, 33], [1189, 47, 17, 30], [1205, 41, 7, 34], [1205, 49, 23, 26], [1220, 42, 8, 34], [1220, 46, 14, 32], [1241, 39, 4, 35], [1241, 49, 20, 29], [1250, 40, 5, 35], [1250, 48, 17, 31], [1258, 46, 13, 33], [1258, 50, 23, 27], [1261, 41, 6, 35], [1261, 49, 19, 30], [1285, 47, 14, 33], [1285, 49, 18, 31], [1300, 38, 2, 36], [1300, 46, 12, 34], [1300, 50, 20, 30], [1305, 39, 3, 36], [1305, 51, 24, 27], [1313, 49, 17, 32], [1313, 51, 23, 28], [1325, 45, 10, 35], [1325, 47, 13, 34], [1325, 51, 22, 29], [1345, 43, 7, 36], [1345, 49, 16, 33], [1360, 44, 8, 36], [1360, 52, 24, 28], [1370, 38, 1, 37], [1370, 52, 23, 29], [1378, 40, 3, 37], [1378, 50, 17, 33], [1385, 41, 4, 37], [1385, 51, 19, 32], [1394, 42, 5, 37], [1394, 48, 13, 35], [1405, 43, 6, 37], [1405, 53, 26, 27], [1417, 47, 11, 36], [1417, 53, 24, 29], [1445, 39, 1, 38], [1445, 51, 17, 34], [1445, 53, 22, 31], [1450, 46, 9, 37], [1450, 50, 15, 35], [1450, 52, 19, 33], [1460, 42, 4, 38], [1460, 54, 26, 28], [1465, 49, 13, 36], [1465, 53, 21, 32], [1469, 43, 5, 38], [1469, 47, 10, 37], [1480, 44, 6, 38], [1480, 52, 18, 34], [1490, 48, 11, 37], [1490, 54, 23, 31], [1508, 46, 8, 38], [1508, 54, 22, 32], [1513, 49, 12, 37], [1513, 55, 27, 28], [1517, 53, 19, 34], [1517, 55, 26, 29], [1525, 41, 2, 39], [1525, 47, 9, 38], [1525, 55, 25, 30], [1530, 42, 3, 39], [1530, 54, 21, 33], [1537, 43, 4, 39], [1537, 55, 24, 31], [1565, 49, 11, 38], [1565, 51, 14, 37], [1570, 46, 7, 39], [1570, 56, 27, 29], [1585, 47, 8, 39], [1585, 53, 17, 36], [1586, 54, 19, 35], [1586, 56, 25, 31], [1625, 45, 5, 40], [1625, 53, 16, 37], [1625, 55, 20, 35], [1625, 57, 28, 29], [1640, 52, 14, 38], [1640, 56, 22, 34], [1649, 47, 7, 40], [1649, 57, 25, 32], [1665, 51, 12, 39], [1665, 57, 24, 33], [1685, 43, 2, 41], [1685, 57, 23, 34], [1690, 44, 3, 41], [1690, 52, 13, 39], [1690, 58, 27, 31], [1700, 50, 10, 40], [1700, 54, 16, 38], [1700, 58, 26, 32], [1717, 47, 6, 41], [1717, 53, 14, 39], [1730, 48, 7, 41], [1730, 56, 19, 37], [1745, 49, 8, 41], [1745, 59, 28, 31], [1765, 43, 1, 42], [1765, 59, 26, 33], [1768, 44, 2, 42], [1768, 56, 18, 38], [1769, 53, 13, 40], [1769, 57, 20, 37], [1780, 46, 4, 42], [1780, 58, 22, 36], [1781, 51, 10, 41], [1781, 59, 25, 34], [1802, 52, 11, 41], [1802, 60, 29, 31], [1810, 56, 17, 39], [1810, 58, 21, 37], [1825, 53, 12, 41], [1825, 55, 15, 40], [1825, 59, 23, 36], [1845, 51, 9, 42], [1845, 57, 18, 39], [1850, 44, 1, 43], [1850, 54, 13, 41], [1850, 60, 25, 35], [1853, 45, 2, 43], [1853, 59, 22, 37], [1865, 47, 4, 43], [1865, 61, 29, 32], [1885, 49, 6, 43], [1885, 53, 11, 42], [1885, 59, 21, 38], [1885, 61, 27, 34], [1898, 50, 7, 43], [1898, 60, 23, 37], [1921, 59, 20, 39], [1921, 61, 25, 36], [1924, 58, 18, 40], [1924, 62, 30, 32], [1930, 52, 9, 43], [1930, 62, 29, 33], [1937, 45, 1, 44], [1937, 57, 16, 41], [1940, 46, 2, 44], [1940, 62, 28, 34], [1945, 47, 3, 44], [1945, 61, 24, 37], [1961, 49, 5, 44], [1961, 59, 19, 40], [1970, 54, 11, 43], [1970, 58, 17, 41], [1972, 50, 6, 44], [1972, 62, 26, 36], [1985, 51, 7, 44], [1985, 63, 31, 32], [1989, 57, 15, 42], [1989, 63, 30, 33], [2000, 52, 8, 44], [2000, 60, 20, 40], [2005, 59, 18, 41], [2005, 61, 22, 39], [2020, 58, 16, 42], [2020, 62, 24, 38], [2041, 49, 4, 45], [2041, 61, 21, 40], [2045, 57, 14, 43], [2045, 63, 26, 37], [2050, 50, 5, 45], [2050, 62, 23, 39], [2050, 64, 31, 33], [2074, 52, 7, 45], [2074, 58, 15, 43], [2080, 56, 12, 44], [2080, 64, 28, 36], [2105, 57, 13, 44], [2105, 59, 16, 43], [2117, 47, 1, 46], [2117, 65, 31, 34], [2120, 48, 2, 46], [2120, 64, 26, 38], [2125, 49, 3, 46], [2125, 55, 10, 45], [2125, 61, 19, 42], [2125, 65, 30, 35], [2132, 50, 4, 46], [2132, 58, 14, 44], [2146, 56, 11, 45], [2146, 64, 25, 39], [2165, 53, 7, 46], [2165, 63, 22, 41], [2173, 61, 18, 43], [2173, 65, 27, 38], [2180, 54, 8, 46], [2180, 66, 32, 34], [2197, 55, 9, 46], [2197, 65, 26, 39], [2210, 48, 1, 47], [2210, 62, 19, 43], [2210, 64, 23, 41], [2210, 66, 29, 37], [2225, 51, 4, 47], [2225, 61, 17, 44], [2225, 65, 25, 40], [2245, 53, 6, 47], [2245, 67, 33, 34], [2249, 63, 20, 43], [2249, 67, 32, 35], [2250, 60, 15, 45], [2250, 66, 27, 39], [2257, 65, 24, 41], [2257, 67, 31, 36], [2260, 58, 12, 46], [2260, 62, 18, 44], [2285, 59, 13, 46], [2285, 67, 29, 38], [2290, 56, 9, 47], [2290, 64, 21, 43], [2305, 49, 1, 48], [2305, 67, 28, 39], [2314, 62, 17, 45], [2314, 68, 33, 35], [2320, 52, 4, 48], [2320, 68, 32, 36], [2329, 53, 5, 48], [2329, 67, 27, 40], [2330, 58, 11, 47], [2330, 68, 31, 37], [2340, 54, 6, 48], [2340, 66, 24, 42], [2353, 55, 7, 48], [2353, 59, 12, 47], [2378, 60, 13, 47], [2378, 66, 23, 43], [2385, 57, 9, 48], [2385, 69, 33, 36], [2405, 51, 2, 49], [2405, 61, 14, 47], [2405, 63, 17, 46], [2405, 69, 31, 38], [2410, 52, 3, 49], [2410, 68, 27, 41], [2425, 59, 11, 48], [2425, 65, 20, 45], [2425, 67, 24, 43], [2440, 64, 18, 46], [2440, 68, 26, 42], [2465, 57, 8, 49], [2465, 63, 16, 47], [2465, 67, 23, 44], [2465, 69, 28, 41], [2482, 58, 9, 49], [2482, 70, 31, 39], [2500, 62, 14, 48], [2500, 70, 30, 40], [2501, 51, 1, 50], [2501, 59, 10, 49], [2509, 53, 3, 50], [2509, 67, 22, 45], [2516, 54, 4, 50], [2516, 66, 20, 46], [2522, 60, 11, 49], [2522, 70, 29, 41], [2525, 55, 5, 50], [2525, 69, 26, 43], [2525, 71, 34, 37], [2533, 65, 18, 47], [2533, 71, 33, 38], [2545, 61, 12, 49], [2545, 71, 32, 39], [2561, 69, 25, 44], [2561, 71, 31, 40], [2570, 62, 13, 49], [2570, 66, 19, 47], [2581, 59, 9, 50], [2581, 71, 30, 41], [2600, 60, 10, 50], [2600, 68, 22, 46], [2600, 72, 34, 38], [2605, 53, 2, 51], [2605, 71, 29, 42], [2610, 54, 3, 51], [2610, 72, 33, 39], [2626, 56, 5, 51], [2626, 64, 15, 49], [2650, 58, 7, 51], [2650, 68, 21, 47], [2650, 70, 25, 45], [2665, 59, 8, 51], [2665, 67, 19, 48], [2665, 71, 27, 44], [2665, 73, 36, 37], [2669, 63, 13, 50], [2669, 73, 35, 38], [2690, 66, 17, 49], [2690, 72, 29, 43], [2701, 61, 10, 51], [2701, 71, 26, 45], [2705, 53, 1, 52], [2705, 73, 32, 41], [2720, 56, 4, 52], [2720, 72, 28, 44], [2725, 65, 15, 50], [2725, 67, 18, 49], [2725, 73, 31, 42], [2740, 58, 6, 52], [2740, 74, 36, 38], [2745, 63, 12, 51], [2745, 69, 21, 48], [2756, 66, 16, 50], [2756, 74, 34, 40], [2770, 64, 13, 51], [2770, 74, 33, 41], [2785, 61, 9, 52], [2785, 71, 24, 47], [2788, 70, 22, 48], [2788, 74, 32, 42], [2810, 54, 1, 53], [2810, 74, 31, 43], [2813, 55, 2, 53], [2813, 75, 37, 38], [2825, 57, 4, 53], [2825, 63, 11, 52], [2825, 75, 35, 40], [2834, 58, 5, 53], [2834, 72, 25, 47], [2845, 59, 6, 53], [2845, 73, 27, 46], [2873, 61, 8, 53], [2873, 65, 13, 52], [2873, 75, 32, 43], [2885, 71, 22, 49], [2885, 73, 26, 47], [2890, 62, 9, 53], [2890, 68, 17, 51], [2890, 76, 37, 39], [2900, 66, 14, 52], [2900, 70, 20, 50], [2900, 74, 28, 46], [2920, 56, 2, 54], [2920, 76, 34, 42], [2925, 57, 3, 54], [2925, 69, 18, 51], [2925, 75, 30, 45], [2929, 67, 15, 52], [2929, 73, 25, 48], [2930, 64, 11, 53], [2930, 72, 23, 49], 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[9412, 110, 14, 96], [9412, 118, 24, 94], [9418, 100, 3, 97], [9418, 130, 43, 87], [9425, 101, 4, 97], [9425, 115, 20, 95], [9425, 123, 31, 92], [9425, 129, 41, 88], [9425, 135, 55, 80], [9425, 137, 64, 73], [9434, 102, 5, 97], [9434, 132, 47, 85], [9445, 103, 6, 97], [9445, 137, 63, 74], [9469, 127, 37, 90], [9469, 137, 62, 75], [9490, 106, 9, 97], [9490, 122, 29, 93], [9490, 134, 51, 83], [9490, 136, 57, 79], [9505, 113, 17, 96], [9505, 131, 44, 87], [9509, 107, 10, 97], [9509, 117, 22, 95], [9512, 120, 26, 94], [9512, 132, 46, 86], [9529, 133, 48, 85], [9529, 137, 60, 77], [9530, 108, 11, 97], [9530, 138, 67, 71], [9540, 114, 18, 96], [9540, 138, 66, 72], [9553, 109, 12, 97], [9553, 125, 33, 92], [9554, 118, 23, 95], [9554, 138, 65, 73], [9565, 121, 27, 94], [9565, 137, 59, 78], [9577, 115, 19, 96], [9577, 127, 36, 91], [9593, 131, 43, 88], [9593, 135, 52, 83], [9594, 132, 45, 87], [9594, 138, 63, 75], [9605, 99, 1, 98], [9605, 111, 14, 97], [9605, 133, 47, 86], [9605, 137, 58, 79], [9620, 102, 4, 98], [9620, 122, 28, 94], [9620, 126, 34, 92], [9620, 138, 62, 76], [9640, 104, 6, 98], [9640, 136, 54, 82], [9650, 120, 25, 95], [9650, 128, 37, 91], [9650, 138, 61, 77], [9657, 117, 21, 96], [9657, 135, 51, 84], [9665, 113, 16, 97], [9665, 139, 68, 71], [9673, 125, 32, 93], [9673, 139, 67, 72], [9685, 107, 9, 98], [9685, 131, 42, 89], [9685, 133, 46, 87], [9685, 139, 66, 73], [9698, 114, 17, 97], [9698, 136, 53, 83], [9700, 118, 22, 96], [9700, 130, 40, 90], [9700, 134, 48, 86], [9701, 121, 26, 95], [9701, 139, 65, 74], [9725, 109, 11, 98], [9725, 129, 38, 91], [9725, 135, 50, 85], [9745, 119, 23, 96], [9745, 139, 63, 76], [9760, 128, 36, 92], [9760, 136, 52, 84], [9770, 116, 19, 97], [9770, 132, 43, 89], [9773, 111, 13, 98], [9773, 139, 62, 77], [9797, 125, 31, 94], [9797, 135, 49, 86], [9802, 100, 1, 99], [9802, 130, 39, 91], [9802, 140, 69, 71], [9805, 101, 2, 99], [9805, 127, 34, 93], [9805, 137, 54, 83], [9805, 139, 61, 78], [9809, 117, 20, 97], [9809, 123, 28, 95], [9810, 102, 3, 99], [9810, 138, 57, 81], [9826, 104, 5, 99], [9826, 136, 51, 85], [9841, 121, 25, 96], [9841, 139, 60, 79], [9850, 106, 7, 99], [9850, 118, 21, 97], [9850, 140, 65, 75], [9860, 114, 16, 98], [9860, 126, 32, 94], [9860, 134, 46, 88], [9860, 138, 56, 82], [9865, 107, 8, 99], [9865, 137, 53, 84], [9881, 131, 40, 91], [9881, 139, 59, 80], [9893, 115, 17, 98], [9893, 119, 22, 97], [9925, 125, 30, 95], [9925, 127, 33, 94], [9925, 139, 58, 81], [9928, 116, 18, 98], [9928, 140, 62, 78], [9945, 111, 12, 99], [9945, 123, 27, 96], [9945, 129, 36, 93], [9945, 141, 69, 72], [9953, 135, 47, 88], [9953, 141, 68, 73], [9962, 132, 41, 91], [9962, 140, 61, 79], [9965, 117, 19, 98], [9965, 141, 67, 74], [9970, 112, 13, 99], [9970, 136, 49, 87], [9985, 121, 24, 97], [9985, 131, 39, 92], [9997, 113, 14, 99], [9997, 137, 51, 86], [10000, 124, 28, 96], [10000, 140, 60, 80], [10025, 139, 56, 83], [10025, 141, 64, 77], [10034, 122, 25, 97], [10034, 138, 53, 85], [10045, 119, 21, 98], [10045, 133, 42, 91], [10057, 115, 16, 99], [10057, 125, 29, 96], [10085, 123, 26, 97], [10085, 141, 62, 79], [10088, 120, 22, 98], [10088, 140, 58, 82], [10090, 116, 17, 99], [10090, 142, 69, 73], [10100, 138, 52, 86], [10100, 142, 68, 74], [10114, 128, 33, 95], [10114, 142, 67, 75], [10125, 117, 18, 99], [10125, 135, 45, 90], [10130, 134, 43, 91], [10130, 136, 47, 89], [10132, 130, 36, 94], [10132, 142, 66, 76], [10138, 124, 27, 97], [10138, 140, 57, 83], [10145, 133, 41, 92], [10145, 137, 49, 88], [10170, 132, 39, 93], [10170, 138, 51, 87], [10180, 122, 24, 98], [10180, 142, 64, 78], [10205, 131, 37, 94], [10205, 139, 53, 86], [10205, 141, 59, 82], [10225, 137, 48, 89], [10225, 143, 71, 72], [10229, 123, 25, 98], [10229, 143, 70, 73], [10244, 138, 50, 88], [10244, 142, 62, 80], [10249, 133, 40, 93], [10249, 143, 68, 75], [10250, 126, 29, 97], [10250, 130, 35, 95], [10250, 140, 55, 85], [10280, 124, 26, 98], [10280, 132, 38, 94], [10285, 121, 22, 99], [10285, 143, 66, 77], [10305, 129, 33, 96], [10305, 141, 57, 84], [10309, 127, 30, 97], [10309, 137, 47, 90], [10309, 143, 65, 78], [10330, 122, 23, 99], [10330, 134, 41, 93], [10370, 128, 31, 97], [10370, 142, 59, 83], [10370, 144, 71, 73], [10400, 136, 44, 92], [10400, 144, 68, 76], [10421, 139, 50, 89], [10421, 141, 55, 86], [10445, 127, 29, 98], [10445, 143, 61, 82], [10489, 137, 45, 92], [10489, 143, 60, 83], [10498, 130, 33, 97], [10498, 136, 43, 93], [10517, 135, 41, 94], [10517, 145, 71, 74], [10530, 126, 27, 99], [10530, 144, 63, 81], [10537, 143, 59, 84], [10537, 145, 69, 76], [10553, 141, 53, 88], [10553, 145, 68, 77], [10565, 129, 31, 98], [10565, 131, 34, 97], [10585, 127, 28, 99], [10585, 133, 37, 96], [10585, 137, 44, 93], [10585, 139, 48, 91], [10600, 136, 42, 94], [10600, 140, 50, 90], [10625, 135, 40, 95], [10625, 141, 52, 89], [10625, 145, 65, 80], [10660, 134, 38, 96], [10660, 142, 54, 88], [10660, 146, 72, 74], [10693, 131, 33, 98], [10693, 145, 63, 82], [10701, 129, 30, 99], [10701, 141, 51, 90], [10705, 133, 36, 97], [10705, 143, 56, 87], [10706, 136, 41, 95], [10706, 144, 59, 85], [10730, 142, 53, 89], [10730, 146, 67, 79], [10760, 132, 34, 98], [10760, 144, 58, 86], [10825, 131, 32, 99], [10825, 145, 60, 85], [10829, 133, 35, 98], [10829, 147, 70, 77], [10858, 140, 47, 93], [10858, 146, 63, 83], [10865, 141, 49, 92], [10865, 147, 68, 79], [10900, 134, 36, 98], [10900, 146, 62, 84], [10946, 144, 55, 89], [10946, 146, 61, 85], [10985, 143, 52, 91], [10985, 147, 64, 83], [11050, 140, 45, 95], [11050, 142, 49, 93], [11050, 146, 59, 87], [11050, 148, 67, 81], [11065, 139, 43, 96], [11065, 143, 51, 92], [11090, 138, 41, 97], [11090, 144, 53, 91], [11125, 137, 39, 98], [11125, 145, 55, 90], [11125, 149, 71, 78], [11141, 141, 46, 95], [11141, 149, 70, 79], [11152, 140, 44, 96], [11152, 148, 64, 84], [11170, 136, 37, 99], [11170, 146, 57, 89], [11245, 137, 38, 99], [11245, 149, 66, 83], [11258, 140, 43, 97], [11258, 150, 73, 77], [11285, 139, 41, 98], [11285, 147, 58, 89], [11322, 138, 39, 99], [11322, 150, 69, 81], [11401, 139, 40, 99], [11401, 151, 75, 76], [11413, 149, 62, 87], [11413, 151, 73, 78], [11425, 143, 47, 96], [11425, 151, 72, 79], [11525, 143, 46, 97], [11525, 145, 50, 95], [11540, 142, 44, 98], [11540, 146, 52, 94], [11565, 141, 42, 99], [11565, 147, 54, 93], [11581, 149, 59, 90], [11581, 151, 66, 85], [11645, 147, 53, 94], [11645, 149, 58, 91], [11650, 142, 43, 99], [11650, 152, 69, 83], [11713, 145, 48, 97], [11713, 149, 57, 92], [11713, 151, 63, 88], [11729, 147, 52, 95], [11729, 153, 73, 80], [11752, 148, 54, 94], [11752, 152, 66, 86], [11765, 151, 62, 89], [11765, 153, 71, 82], [11817, 147, 51, 96], [11817, 153, 69, 84], [11890, 152, 63, 89], [11890, 154, 73, 81], [11908, 146, 48, 98], [11908, 154, 72, 82], [12010, 146, 47, 99], [12010, 148, 51, 97], [12013, 151, 58, 93], [12013, 155, 77, 78], [12017, 153, 64, 89], [12017, 155, 76, 79], [12025, 149, 53, 96], [12025, 155, 75, 80], [12125, 153, 62, 91], [12125, 155, 70, 85], [12200, 152, 58, 94], [12200, 156, 74, 82], [12218, 150, 53, 97], [12218, 156, 73, 83], [12308, 150, 52, 98], [12308, 154, 62, 92], [12325, 151, 54, 97], [12325, 155, 65, 90], [12325, 157, 78, 79], [12505, 151, 52, 99], [12505, 157, 69, 88], [12506, 154, 59, 95], [12506, 156, 65, 91], [12545, 153, 56, 97], [12545, 157, 68, 89], [12580, 154, 58, 96], [12580, 158, 72, 86], [12610, 152, 53, 99], [12610, 158, 71, 87], [12937, 155, 56, 99], [12937, 157, 61, 96], [13130, 158, 61, 97], [13130, 162, 79, 83], [13250, 160, 65, 95], [13250, 162, 73, 89], [13325, 159, 61, 98], [13325, 161, 67, 94], [13325, 163, 77, 86], [13505, 161, 64, 97], [13505, 163, 71, 92], [13786, 164, 69, 95], [13786, 166, 81, 85], [14365, 167, 69, 98], [14365, 169, 78, 91], [14425, 167, 68, 99], [14425, 169, 76, 93], [14645, 169, 71, 98], [14645, 171, 82, 89], [14842, 170, 71, 99], [14842, 172, 81, 91]

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This is a good one. I think I'm getting closer, but got stuck at the end.

First exchange: A doesn't know. B also doesn't know, meaning that B's SS must be a number that have more than one combination of perfect squares. I did the same thing as mmiguel1, but continued through 15^2, giving the options: 65 (1,8 or 4,7; sums=9 or 11) 85 (2,9 or 6,7; sums=11 or 13) 125 (2,11 or 5,10; sums=13 or 15) 130 (3,11 or 7,9; sums=14 or 16) 145 (1,12 or 8,9; sums=13 or 17) 170 (1,13 or 7,11; sums=14 or 18) 250 (5,15 or 9,13; sums=20 or 22) 306 (9,15 or 10, 14; sum has to be 24 either way) Second exchange: A knows that B's number must have more than one possible combination of perfect squares. If A's number were 9, 15, 16, 17, 18, 20, or 22, then A would know the answer But A doesn't know this. B realizes that if A doesn't know, then A's number must be 11, 13, 14, or 24 (or maybe a higher number, I'm too lazy to go higher). B still doesn't know the answer. Third exchange: A knows that B realizes the number has to be one of those numbers, but still didn't know the answer. So, B's number can't be 65 or 125 or 130 or 145 or 170 or 250, because B (knowing B's own number and that A's number has to be 11, 13, 14, or 24) doesn't know the answer. But B still doesn't know which one. Fourth exchange: A knows that B doesn't know the answer, but that B knows it must be 11 (2,9), 13 (6,7), or 24(9,15, or 10, 14). A knows the sum, but still doesn't know the answer. B knows then that A's number has to be 24, but I'm stumped as to how B knows which numbers to pick. Or maybe I'm really off track here.

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There are two solutions I could find.

(8,9) and (73,89)

I assumed all integers were less than 100.

with x <= y

from now on sum means x+y and square sum means x^2 + y^2

--------------------------------------------

Starting from basic pairs of numbers from 1 to 100, I found all non-unique sums (basically everything except 2)

and removed the unique sums.

That lists represents the possibilities B can choose from assuming A does not yet know what the answer is.

Explanation: if a sum were unique in the list, then A would know the answer if A had that number. Since A doesn't yet know the answer, A must not have that number.

From that list, I found all non-unique square sums. This list represents what A can choose from at this stage assuming B does not yet know what the answer is. By removing the unique square sums, I ensure that all that remains in the list are numbers where B cannot have determined the answer at this stage.

This was the first iteration (A tries, then B tries)

After 4 iterations of going back and forth, I looked at the list that B could choose from assuming A hasn't figured it out yet.

I then found all the UNIQUE square sums. At this point in the riddle, B has figured out his number. Therefore he must have a square sum number unique in the list.

There were only two: 145, and 13250

I think based on this logic, either of my two solutions would work to solve the puzzle.

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The presentation of this puzzle is very similar to the Sum-Sum of Squares Puzzle presented by *Richard I. Hess in Puzzles from around the world, 1997, Problem 41. The problem can also be found in 1983 issue of Mathematics Magazine, 56, p.177, Problem 1173.

A problem with this presentation is that no upperbound is given. Without an upperbound we can not deduce the numbers the logicians would deduce. In Hess's version, the upperbound given is ≤ 50. The upperbound could have been as high as 75, possibly higher, but around 80 the logicians would be unable to solve it within their conversation length.

Besides a given upperbound limit, in Hess's version Logician B begins the conversation. I have not determined if this makes a difference to who first deduces the number, but as in this puzzle, Logician B is given as the solver. The solution to Hess's presentation is the pair of numbers 8 and 9.

There are many pairs of numbers that exist where the solution can not be logically reasoned to be the number, this is in itself lets us know that the solution must be a number able to be excluded. The last single excluded pair of positive integers where the upperbound is ≤ 75, is 8 and 9.

* I have not confirmed that the actual presentation of the puzzle referenced is as given by Hess, but my analysis of the differences is based from the December 1997 article, "The Impossible Problem", by Torsten Sillke in http://www.math.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product, as well as, my own testing for the solution via a computer program in my search for an upperbound.

Edited by Dej Mar
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The presentation of this puzzle is very similar to the Sum-Sum of Squares Puzzle presented by *Richard I. Hess in Puzzles from around the world, 1997, Problem 41. The problem can also be found in 1983 issue of Mathematics Magazine, 56, p.177, Problem 1173.

A problem with this presentation is that no upperbound is given. Without an upperbound we can not deduce the numbers the logicians would deduce. In Hess's version, the upperbound given is ≤ 50. The upperbound could have been as high as 75, possibly higher, but around 80 the logicians would be unable to solve it within their conversation length.

Besides a given upperbound limit, in Hess's version Logician B begins the conversation. I have not determined if this makes a difference to who first deduces the number, but as in this puzzle, Logician B is given as the solver. The solution to Hess's presentation is the pair of numbers 8 and 9.

There are many pairs of numbers that exist where the solution can not be logically reasoned to be the number, this is in itself lets us know that the solution must be a number able to be excluded. The last single excluded pair of positive integers where the upperbound is ≤ 75, is 8 and 9.

* I have not confirmed that the actual presentation of the puzzle referenced is as given by Hess, but my analysis of the differences is based from the December 1997 article, "The Impossible Problem", by Torsten Sillke in http://www.math.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product, as well as, my own testing for the solution via a computer program in my search for an upperbound.

Great point.

There are multiple pairs of numbers upon which the same "story" depicted in the puzzle could unfold.

Logician B would be able to figure out which pair of these is the answer because he knows the sum of the squares of the numbers.

You and I however cannot figure out which pair Logician B has deduced.

With an arbitrary upper bound, I found two such pairs.

I am sure if I were to increase the upper bound, I could find more.

For doing this problem by hand without a computer program, I would suspect that most people would only come up with the first such pair of numbers if any.

I concur that an upper limit should be given so that the problem has a single, unique answer.

Edited by mmiguel1
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The presentation of this puzzle is very similar to the Sum-Sum of Squares Puzzle presented by *Richard I. Hess in Puzzles from around the world, 1997, Problem 41. The problem can also be found in 1983 issue of Mathematics Magazine, 56, p.177, Problem 1173.

A problem with this presentation is that no upperbound is given. Without an upperbound we can not deduce the numbers the logicians would deduce. In Hess's version, the upperbound given is ≤ 50. The upperbound could have been as high as 75, possibly higher, but around 80 the logicians would be unable to solve it within their conversation length.

Besides a given upperbound limit, in Hess's version Logician B begins the conversation. I have not determined if this makes a difference to who first deduces the number, but as in this puzzle, Logician B is given as the solver. The solution to Hess's presentation is the pair of numbers 8 and 9.

There are many pairs of numbers that exist where the solution can not be logically reasoned to be the number, this is in itself lets us know that the solution must be a number able to be excluded. The last single excluded pair of positive integers where the upperbound is ≤ 75, is 8 and 9.

* I have not confirmed that the actual presentation of the puzzle referenced is as given by Hess, but my analysis of the differences is based from the December 1997 article, "The Impossible Problem", by Torsten Sillke in http://www.math.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product, as well as, my own testing for the solution via a computer program in my search for an upperbound.

The friend of mine who gave me this problem has never failed me before. He is simply the most brilliant problem solver I know. His Dad, a game theory professor gave him the problem over 20 years ago. He freely admitted he couldn't prove there was only one solution as he hadn't used a computer to help him solve it.

I found a file on the internet that listed various numbers that are the sum of more than 2 different pairs of squares. From there, I did it by hand and found 8 and 9 to be the answer. Clearly, there should be an upper bound, if for no other reason than to make it easily provable that there is an actual answer.

Fun puzzle though.

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The presentation of this puzzle is very similar to the Sum-Sum of Squares Puzzle presented by *Richard I. Hess in Puzzles from around the world, 1997, Problem 41. The problem can also be found in 1983 issue of Mathematics Magazine, 56, p.177, Problem 1173.

A problem with this presentation is that no upperbound is given. Without an upperbound we can not deduce the numbers the logicians would deduce. In Hess's version, the upperbound given is ≤ 50. The upperbound could have been as high as 75, possibly higher, but around 80 the logicians would be unable to solve it within their conversation length.

Besides a given upperbound limit, in Hess's version Logician B begins the conversation. I have not determined if this makes a difference to who first deduces the number, but as in this puzzle, Logician B is given as the solver. The solution to Hess's presentation is the pair of numbers 8 and 9.

There are many pairs of numbers that exist where the solution can not be logically reasoned to be the number, this is in itself lets us know that the solution must be a number able to be excluded. The last single excluded pair of positive integers where the upperbound is ≤ 75, is 8 and 9.

* I have not confirmed that the actual presentation of the puzzle referenced is as given by Hess, but my analysis of the differences is based from the December 1997 article, "The Impossible Problem", by Torsten Sillke in http://www.math.uni-bielefeld.de/~sillke/PUZZLES/logic_sum_product, as well as, my own testing for the solution via a computer program in my search for an upperbound.

Well, I just had an interesting conversation with my friend. He stands by his problem and he is working on a proof that 8 and 9 is the only answer to his puzzle. His father, the game theory professor, showed him a proof, but my friend is not sure it's airtight. So he's working on something better.

This problem raises many interesting issues, where "bounds" are concerned. My friend is convinced that the reason mmiguel found the alternative answer of 73,89 is precisely because he admitted that he "assumed" the numbers were under 100. This changes the puzzle in ways that have tripped up famous puzzle masters in unexpected ways before. If you think about it, the puzzle, as constructed by my friend, has a lower bound of 0. It is precisely this bound that leads the logicians to have their conversation with B finally declaring he knows the numbers (BTW, Logicians A's next statement would be "oh, so now that you know the numbers, I know the numbers!).

If you assume the numbers mut be under 100, then as you try to solve the puzzle and get inside the heads of the perfect logicians, you must also assume that they are aware of the upper bound as well. There are only so many combinations using numbers near 100, so if the logician A was given a sum of around 162 and logician B was given a sum of their squares of around 13,250, knowing the upper bound for each number is 100, makes it possible for the logicians to end the conversation on the 7th exchange with the conclusion that the numbers are 73 and 89. But if there wasn't an upper bound, that might not be possible.

You mention that the Hess puzzle has an upper bound of 50, and it might certainly be possible that with this specific upper bound, the only answer is 8 and 9. But, that doesn't mean the puzzle is better with an upper bound. It just might be easier. The really interesting task is to solve the puzzle as stated, and then prove that 8 and 9 are the only possible solutions. My friend is trying to do just that and I believe he will succeed. That proof is well beyond my mathematical capabilities.

Earlier in this spoiler, I mentioned that the issue of "bounds" as tripped up even the best puzzle masters. The particular case I am referring to is none other than Martin Gardner, the legendary puzzle master for Scientific American. The puzzle mentioned in the link I provide below is quite fun and I posted it here last year. It has an upper bound of 100. And, though difficult, it's quite possible to solve. Gardner decided he wanted to make it "easier" for his readers and put an upper bound of 20, as the solution was within that bound. But in so doing, he actually made the puzzle impossible to solve. He admitted his mistake in a later issue of the magazine. You will see the discussion of the puzzle, and then of his mistake about half way down the link. I find this to be very interesting stuff.

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Well, I just had an interesting conversation with my friend. He stands by his problem and he is working on a proof that 8 and 9 is the only answer to his puzzle. His father, the game theory professor, showed him a proof, but my friend is not sure it's airtight. So he's working on something better.

This problem raises many interesting issues, where "bounds" are concerned. My friend is convinced that the reason mmiguel found the alternative answer of 73,89 is precisely because he admitted that he "assumed" the numbers were under 100. This changes the puzzle in ways that have tripped up famous puzzle masters in unexpected ways before. If you think about it, the puzzle, as constructed by my friend, has a lower bound of 0. It is precisely this bound that leads the logicians to have their conversation with B finally declaring he knows the numbers (BTW, Logicians A's next statement would be "oh, so now that you know the numbers, I know the numbers!).

If you assume the numbers mut be under 100, then as you try to solve the puzzle and get inside the heads of the perfect logicians, you must also assume that they are aware of the upper bound as well. There are only so many combinations using numbers near 100, so if the logician A was given a sum of around 162 and logician B was given a sum of their squares of around 13,250, knowing the upper bound for each number is 100, makes it possible for the logicians to end the conversation on the 7th exchange with the conclusion that the numbers are 73 and 89. But if there wasn't an upper bound, that might not be possible.

You mention that the Hess puzzle has an upper bound of 50, and it might certainly be possible that with this specific upper bound, the only answer is 8 and 9. But, that doesn't mean the puzzle is better with an upper bound. It just might be easier. The really interesting task is to solve the puzzle as stated, and then prove that 8 and 9 are the only possible solutions. My friend is trying to do just that and I believe he will succeed. That proof is well beyond my mathematical capabilities.

Earlier in this spoiler, I mentioned that the issue of "bounds" as tripped up even the best puzzle masters. The particular case I am referring to is none other than Martin Gardner, the legendary puzzle master for Scientific American. The puzzle mentioned in the link I provide below is quite fun and I posted it here last year. It has an upper bound of 100. And, though difficult, it's quite possible to solve. Gardner decided he wanted to make it "easier" for his readers and put an upper bound of 20, as the solution was within that bound. But in so doing, he actually made the puzzle impossible to solve. He admitted his mistake in a later issue of the magazine. You will see the discussion of the puzzle, and then of his mistake about half way down the link. I find this to be very interesting stuff.

Ah,

so you are saying if I had a higher bound, then that second answer would not be unique after all the iterations.

I can believe that.

The proof sounds difficult.

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Ah,

so you are saying if I had a higher bound, then that second answer would not be unique after all the iterations.

I can believe that.

The proof sounds difficult.

I'm actually saying more than that. I'm saying that if you had a higher bound, you may have found other solutions. I'm also saying that if there is no upper bound, there is most likely 1 solution; the one listed first in your answer (don't want to spoil anything outside a spoiler button).

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